Octonions are not as well known as the quaternions and complex numbers, which are much more widely studied and used. Despite this, they have some interesting properties and are related to a number of exceptional structures in mathematics, among them the exceptional Lie groups. Additionally, octonions have applications in fields such as string theory, special relativity, and quantum logic.

The octonions were discovered in 1843 by John T. Graves, inspired by his friend W. R. Hamilton's discovery of quaternions. Graves called his discovery octaves, and mentioned them in a letter to Hamilton dated 16 December 1843, but his first publication of his result in (Graves 1845) was slightly later than Arthur Cayley's article on them. The octonions were discovered independently by Cayley[1] and are sometimes referred to as Cayley numbers or the Cayley algebra. Hamilton described the early history of Graves' discovery.[2]

Addition and subtraction of octonions is done by adding and subtracting corresponding terms and hence their coefficients, like quaternions. Multiplication is more complex. Multiplication is distributive over addition, so the product of two octonions can be calculated by summing the products of all the terms, again like quaternions. The product of each pair of terms can be given by multiplication of the coefficients and a multiplication table of the unit octonions, like this one: (This table is due to Arthur Cayley (1845) and John T. Graves (1843).)[3]

ej{\displaystyle e_{j}}

eiej{\displaystyle e_{i}e_{j}}

e0{\displaystyle e_{0}}

e1{\displaystyle e_{1}}

e2{\displaystyle e_{2}}

e3{\displaystyle e_{3}}

e4{\displaystyle e_{4}}

e5{\displaystyle e_{5}}

e6{\displaystyle e_{6}}

e7{\displaystyle e_{7}}

ei{\displaystyle e_{i}}

e0{\displaystyle e_{0}}

e0{\displaystyle e_{0}}

e1{\displaystyle e_{1}}

e2{\displaystyle e_{2}}

e3{\displaystyle e_{3}}

e4{\displaystyle e_{4}}

e5{\displaystyle e_{5}}

e6{\displaystyle e_{6}}

e7{\displaystyle e_{7}}

e1{\displaystyle e_{1}}

e1{\displaystyle e_{1}}

−e0{\displaystyle -e_{0}}

e3{\displaystyle e_{3}}

−e2{\displaystyle -e_{2}}

e5{\displaystyle e_{5}}

−e4{\displaystyle -e_{4}}

−e7{\displaystyle -e_{7}}

e6{\displaystyle e_{6}}

e2{\displaystyle e_{2}}

e2{\displaystyle e_{2}}

−e3{\displaystyle -e_{3}}

−e0{\displaystyle -e_{0}}

e1{\displaystyle e_{1}}

e6{\displaystyle e_{6}}

e7{\displaystyle e_{7}}

−e4{\displaystyle -e_{4}}

−e5{\displaystyle -e_{5}}

e3{\displaystyle e_{3}}

e3{\displaystyle e_{3}}

e2{\displaystyle e_{2}}

−e1{\displaystyle -e_{1}}

−e0{\displaystyle -e_{0}}

e7{\displaystyle e_{7}}

−e6{\displaystyle -e_{6}}

e5{\displaystyle e_{5}}

−e4{\displaystyle -e_{4}}

e4{\displaystyle e_{4}}

e4{\displaystyle e_{4}}

−e5{\displaystyle -e_{5}}

−e6{\displaystyle -e_{6}}

−e7{\displaystyle -e_{7}}

−e0{\displaystyle -e_{0}}

e1{\displaystyle e_{1}}

e2{\displaystyle e_{2}}

e3{\displaystyle e_{3}}

e5{\displaystyle e_{5}}

e5{\displaystyle e_{5}}

e4{\displaystyle e_{4}}

−e7{\displaystyle -e_{7}}

e6{\displaystyle e_{6}}

−e1{\displaystyle -e_{1}}

−e0{\displaystyle -e_{0}}

−e3{\displaystyle -e_{3}}

e2{\displaystyle e_{2}}

e6{\displaystyle e_{6}}

e6{\displaystyle e_{6}}

e7{\displaystyle e_{7}}

e4{\displaystyle e_{4}}

−e5{\displaystyle -e_{5}}

−e2{\displaystyle -e_{2}}

e3{\displaystyle e_{3}}

−e0{\displaystyle -e_{0}}

−e1{\displaystyle -e_{1}}

e7{\displaystyle e_{7}}

e7{\displaystyle e_{7}}

−e6{\displaystyle -e_{6}}

e5{\displaystyle e_{5}}

e4{\displaystyle e_{4}}

−e3{\displaystyle -e_{3}}

−e2{\displaystyle -e_{2}}

e1{\displaystyle e_{1}}

−e0{\displaystyle -e_{0}}

Most off-diagonal elements of the table are antisymmetric, making it almost a skew-symmetric matrix except for the elements on the main diagonal, as well as the row and column for which e0 is an operand.

The above definition though is not unique, but is only one of 480 possible definitions for octonion multiplication with e0 = 1. The others can be obtained by permuting and changing the signs of the non-scalar basis elements. The 480 different algebras are isomorphic, and there is rarely a need to consider which particular multiplication rule is used. Each of these 480 definitions is invariant up to signs under some 7-cycle of the points (1234567), and for each 7-cycle there are four definitions, differing by signs and reversal of order. A common choice is to use the definition invariant under the 7-cycle (1234567) with e1e2 = e4 as it is particularly easy to remember the multiplication.

A variation of this sometimes used is to label the elements of the basis by the elements ∞, 0, 1, 2, ..., 6, of the projective line over the finite field of order 7. The multiplication is then given by e∞ = 1 and e1e2 = e4, and all expressions obtained from this by adding a constant (mod 7) to all subscripts: in other words using the 7 triples (124) (235) (346) (450) (561) (602) (013). These are the nonzero codewords of the quadratic residue code of length 7 over the Galois field of two elements, GF(2). There is a symmetry of order 7 given by adding a constant mod 7 to all subscripts, and also a symmetry of order 3 given by multiplying all subscripts by one of the quadratic residues 1, 2, 4 mod 7.[5][6]

The multiplication table for a geometric algebra of signature (−−−−) can be given in terms of the following 7 quaternionic triples (omitting the identity element): (I,j,k), (i,J,k), (i,j,K), (I,J,K), (∗I,i,m), (∗J,j,m), (∗K,k,m) in which the lowercase items are vectors (mathematics and physics) and the uppercase ones are bivectors and ∗ = mijk (which is in fact the Hodge dual operator). If the ∗ is forced to be equal to the identity then the multiplication ceases to be associative, but the ∗ may be removed from the multiplication table resulting in an octonion multiplication table.

Note that in keeping ∗ = mijk associative and thus not reducing the 4-dimensional geometric algebra to an octonion one, the whole multiplication table can be derived from the equation for ∗. Consider the gamma matrices. The formula defining the fifth gamma matrix shows that it is the ∗ of a four-dimensional geometric algebra of the gamma matrices.

A more systematic way of defining the octonions is via the Cayley–Dickson construction. Just as quaternions can be defined as pairs of complex numbers, the octonions can be defined as pairs of quaternions. Addition is defined pairwise. The product of two pairs of quaternions (a, b) and (c, d) is defined by

A 3D mnemonic visualization showing the 7 triads as hyperplanes through the Real (e0{\displaystyle e_{0}}) vertex of the octonion example given above.[7]

A convenient mnemonic for remembering the products of unit octonions is given by the diagram at the right, which represents the multiplication table of Cayley and Graves.[3][8] This diagram with seven points and seven lines (the circle through 1, 2, and 3 is considered a line) is called the Fano plane. The lines are oriented. The seven points correspond to the seven standard basis elements of Im(O) (see definition below). Each pair of distinct points lies on a unique line and each line runs through exactly three points.

Let (a, b, c) be an ordered triple of points lying on a given line with the order specified by the direction of the arrow. Then multiplication is given by

The octonions do satisfy a weaker form of associativity: they are alternative. This means that the subalgebra generated by any two elements is associative. Actually, one can show that the subalgebra generated by any two elements of O is isomorphic to R, C, or H, all of which are associative. Because of their non-associativity, octonions do not have matrix representations, unlike quaternions.

The octonions do retain one important property shared by R, C, and H: the norm on O satisfies

The set of all automorphisms of O forms a group called G2.[10] The group G2 is a simply connected, compact, real Lie group of dimension 14. This group is the smallest of the exceptional Lie groups and is isomorphic to the subgroup of Spin(7) that preserves any chosen particular vector in its 8-dimensional real spinor representation. The group Spin(7) is in turn a subgroup of the group of isotopies described below.

An isotopy of an algebra is a triple of bijective linear maps a, b, c such that if xy = z then a(x)b(y) = c(z). For a = b = c this is the same as an automorphism. The isotopy group of an algebra is the group of all isotopies, which contains the group of automorphisms as a subgroup.

The isotopy group of the octonions is the group Spin8(R), with a, b, and c acting as the three 8-dimensional representations.[11] The subgroup of elements where c fixes the identity is the subgroup Spin7(R), and the subgroup where a, b, and c all fix the identity is the automorphism group G2.

There are several natural ways to choose an integral form of the octonions. The simplest is just to take the octonions whose coordinates are integers. This gives a nonassociative algebra over the integers called the Gravesian octonions. However it is not a maximal order (in the sense of ring theory); there are exactly 7 maximal orders containing it. These 7 maximal orders are all equivalent under automorphisms. The phrase "integral octonions" usually refers to a fixed choice of one of these seven orders.

These maximal orders were constructed by Kirmse (1925), Dickson and Bruck as follows. Label the 8 basis vectors by the points of the projective plane over the field with 7 elements. First form the "Kirmse integers" : these consist of octonions whose coordinates are integers or half integers, and that are half integers (that is, halves of odd integers) on one of the 16 sets

of the extended quadratic residue code of length 8 over the field of 2 elements, given by ∅, (∞124) and its images under adding a constant mod 7, and the complements of these 8 sets. Then switch infinity and any one other coordinate; this operation creates a bijection of the Kirmse integers onto a different set, which is a maximal order. There are 7 ways to do this, giving 7 maximal orders, which are all equivalent under cyclic permutations of the 7 coordinates 0123456. (Kirmse incorrectly claimed that the Kirmse integers also form a maximal order, so he thought there were 8 maximal orders rather than 7, but as Coxeter (1946) pointed out they are not closed under multiplication; this mistake occurs in several published papers.)

The Kirmse integers and the 7 maximal orders are all isometric to the E8 lattice rescaled by a factor of 1/√2. In particular there are 240 elements of minimum nonzero norm 1 in each of these orders, forming a Moufang loop of order 240.

The integral octonions have a "division with remainder" property: given integral octonions a and b ≠ 0, we can find q and r with a = qb + r, where the remainder r has norm less than that of b.

In the integral octonions, all left ideals and right ideals are 2-sided ideals, and the only 2-sided ideals are the principal ideals nO where n is a non-negative integer.

The integral octonions have a version of factorization into primes, though it is not straightforward to state because the octonions are not associative so the product of octonions depends on the order in which one does the products. The irreducible integral octonions are exactly those of prime norm, and every integral octonion can be written as a product of irreducible octonions. More precisely an integral octonion of norm mn can be written as a product of integral octonions of norms m and n.

The automorphism group of the integral octonions is the group G2(F2) of order 12096, which has a simple subgroup of index 2 isomorphic to the unitary group 2A2(32). The isotopy group of the integral octonions is the perfect double cover of the group of rotations of the E8 lattice.